# The effect of wall heating on instability of channel flow ― CORRIGENDUM

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TL;DR: In this article, a review highlights the profound and unexpected ways in which viscosity varying in space and time can affect flow and the most striking manifestations are through alterations of flow stability, as established in model shear flows and industrial applications.

Abstract: This review highlights the profound and unexpected ways in which viscosity varying in space and time can affect flow. The most striking manifestations are through alterations of flow stability, as established in model shear flows and industrial applications. Future studies are needed to address the important effect of viscosity stratification in such diverse environments as Earth's core, the Sun, blood vessels, and the re-entry of spacecraft.

191 citations

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TL;DR: In this paper, the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is investigated and the maximum optimal transient growth Gmax of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr.

Abstract: An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth Gmax of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbers Re, the supremum of Gmax is always attained for streamwise-uniform perturbations (or nearly streamwise-uniform perturbations, in the case of plane Couette flow) which produce large streamwise streaks and Rayleigh-Benard convection rolls (RB). The optimal growth curves retain the same large-Reynolds-number scaling as in pure shear flow. A 3D vector model of the governing equations demonstrates that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this ...

21 citations

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TL;DR: In this article, the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is investigated, showing that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this mechanism is secondary and negligible.

Abstract: An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth G max of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbersRe, the supremum of G max is always attained for streamwise-uniform perturbations (or nearly streamwise-uniform perturbations, in the case of plane Couette flow) which produce large streamwise streaks and Rayleigh-Benard convection rolls (RB). The optimal growth curves retain the same large-Reynolds-number scaling as in pure shear flow. A 3D vector model of the governing equations demonstrates that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this mechanism is secondary and negligible. The optimal input for the largest long-time response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the RBeigenmode without its streamwise velocity component. These short-time and long-time responses depict, to leading order, the optimal transient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. Over a general class of norms that can be considered as growth functions, the results remain qualitatively similar, for example, the dominant adjoint eigenmode still approximates the maximum optimal response.

12 citations

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TL;DR: In this article, the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects is investigated analytically, and the complete inviscid problem is also solved as an initi...

Abstract: We investigate analytically the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects. The complete inviscid problem is also solved as an initi...

7 citations

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TL;DR: In this article, the authors investigated linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point, and showed that the base flow is modally more unstable in the subcritical regime, inviscid unstable in transcritical regime and significantly more stable in the supercritical regime.

Abstract: The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide (CO 2) at a pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number (Re = ρ∗ wu∗ rh∗/μ∗ w), the product of the Prandtl (Pr=μ∗ wC∗ pw/κ∗ w) and Eckert numbers (Ec=u∗ 2 r/C∗ pwT∗ w) and the wall temperature that in addition to pressure determine the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered: the subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical and supercritical temperature regimes. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is modally more unstable in the subcritical regime, inviscid unstable in the transcritical regime and significantly more stable in the supercritical regime. Following the principle of corresponding states, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

4 citations

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TL;DR: In this paper, the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is investigated. But the authors focus on the effects of the wall heating on channel flow.

Abstract: A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Benard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

51 citations

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TL;DR: In this article, the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. But the authors do not consider the effects of wall cooling on channel flow.

Abstract: A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Rayleigh-Benard and Poiseuille modes for Grashof numbers upto $\sim 25000$, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary, destabilising significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

47 citations

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